5. (a) Explain what you understand by a critical region of a test statistic. The number of breakdowns per day in a large fleet of hire cars has a Poisson distribution with mean \(\frac { 1 } { 7 }\).
(b) Find the probability that on a particular day there are fewer than 2 breakdowns.
(c) Find the probability that during a 14-day period there are at most 4 breakdowns. The cars are maintained at a garage. The garage introduced a weekly check to try to decrease the number of cars that break down. In a randomly selected 28-day period after the checks are introduced, only 1 hire car broke down.
(d) Test, at the \(5 \%\) level of significance, whether or not the mean number of breakdowns has decreased. State your hypotheses clearly.